Here, \(u\)  is a column vector of
displacements from lattice equilibrium positions for all the
degrees of freedom, multiplied by the square root of mass; \(p\)  is
the conjugate momentum vector, and \(K\)  is the force constant matrix.
The superscript \(T\)  stands for the matrix transpose. \({ \tilde{A} }\)  is
an antisymmetric real matrix, which is block diagonal with elements
\( \Lambda =\left(\begin{array}{rr} 0 & h \\-h & 0 \\\end{array}\right)\)  (in two dimensions), where \(h\)  is proportional to the magnetic field,
and has the dimension of frequency. For simplicity, we will call \(h\)  magnetic
field later. According to
[1]}, \(h\)  is estimated to be \(0.1\, {\rm cm}^{-1} \approx 3 \times 10^9 \,{ \rm rad\,Hz} \)  at a magnetic field \(\textbf {\emph {B}} = 1\, {\rm T}\)  and a temperature \(T = 5.45\,{\rm K}\) , which
is within the possible range of the coupling strength in ionic insulators [2]}, [3]}. The on-site term, \(u^T{ \tilde{A} }p\) , can be interpreted as the Raman (or spin-phonon) interaction. Based on quantum theory and symmetry consideration, the phenomenological description of the spin-phonon interaction was proposed many years ago [2]}, [3]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}. From the first row
of Eq. (REF ), we find both of the two terms are positive definite,
thus the Hamiltonian  (REF ) is positive definite.
The origin of the Hamiltonian for the PHE is discussed in
detail in the supplementary information of Ref. [12]}.
